These are questions I never got to in an earlier post. Sorry it has taken me time to return to them, though I am hoping some of the footnotes I added to that post may help with these. To answer you more directly, though, I would argue that it depends on what one means by being part of a parish as to whether I answer your first question yes or no. There is the rare situation where a diocesan hermit lives on the premises of a monastery and attends liturgy, and sometimes liturgy of the hours, etc., with the monastic community. Those rare instances aside, most diocesan hermits depend upon the parish for their sacramental life and are a part of the parish in at least that sense. When you ask about not participating in a diocese, the answer is definitely no, because, by definition, a c 603 hermit is consecrated as part of a local (diocesan) Church. She is part of the life of that local Church as well as of the universal Church. This will ordinarily imply being an active member of a parish within that diocese, at least as the source of her sacramental life.
However, some diocesan hermits are involved in the life of the parish in other ways. For instance, I used to do a liturgy of the Word with Communion for the daily Mass group on my pastor's days off. Later, I did that only once or twice a month, and another Sister and lay person took the 2 alternate days, during the month. Once a week, during the school year, I also teach a Scripture class by ZOOM. This is for the parish, but we also have a few people joining us from outside the parish as well. Finally, I do spiritual direction, and while that is open to parishioners, I mainly have clients from outside the parish. So long as a hermit depends on the parish for her sacramental life and contributes even in very limited ways to the life of the parish, especially by being a resource for prayer and for the occasional conversation with parishioners who might want to talk, s/he is an active participant in the parish. I can't see any consecrated Catholic Hermit not participating in parish life at least to the extent of her sacramental life and being a resource for prayer and occasional conversations with those in need. For my comments on Hermits and Eucharistic attendance, please see, Eucharistic Spirituality.Remember that to call oneself a Catholic Hermit is something only the Church herself may permit one to do. After all, to say one is a Catholic Hermit is to say far more than that one is a Catholic and a hermit. It means to live eremitical life as the Church understands it, and to do so in her Name. To be a Catholic means to be baptized and thus commissioned to live the Christian faith in the name of the Catholic Church and in the way she understands and strives to understand and express that faith. Thus, the Catholic laity is given permission at baptism to call themselves Catholic and to strive to live this vocation ever more fully. With other vocations within the Church, priesthood, religious life, consecrated virginity, eremitical life, etc., the Church herself admits candidates to candidacy and a process of mutual discernment. If, through the mediation of the Church, the person is ordained, professed, and/or consecrated by God, they begin to live this specific vocation in the name of the Church and become a Catholic priest, Catholic Sister or Brother, Catholic hermit, and so forth.
The Mystical Body of Christ (or of the Church) refers to the entire Church, on earth and beyond it. What is mystical about it is the way it is composed and held together by God, especially in the Risen Christ and the Holy Spirit. Mystical ordinarily refers to the absolute Mystery of God and to whatever is empowered by that Mystery. It does not refer to one part of the Church, say a "mystical" or "spiritual" part, to the exclusion of the rest of the Church (say, the embodied and very human part). As I noted in my earlier post to you, just as Paul speaks of spiritual people and fleshly people, meaning, respectively, the whole person either under the power of the Spirit or the whole person under the power of Sin, the Mystical Body refers to the whole Church, both on earth and beyond it, under the power of God in the Holy Spirit. The phrase is meant to indicate that what holds the Church together and is the source of its ongoing life is God; it is not simply a large earthly or human organization or institution, nor simply a good idea put forward by human beings who needed a way to worship once a week. It is a privileged way we participate in, experience, and are empowered to help others to experience God's life and sovereignty (God's reign or Kingdom) in our world today. (It is not the Kingdom, but it participates in that Reign of God and helps mediate it to our world.)An ecclesial vocation is similarly distinct from merely being a member of the Church (if one can ever be said to be merely a member of the Church), though it presupposes one is an active member of the Church, yes. Most Catholics live their lives for the sake of the Gospel and do so outside the visible boundaries of the Church. They support the Church with their time, talent, and treasure, as the saying goes; however, their vocations are lived for the sake of their families, and society (school systems, businesses, country, state, county, etc.), and not for the sake of the Church itself. Some vocations, however, don't simply support the Church, and are not merely lived for the sake of the Gospel, as critically important as these things are. These vocations are lived for the sake of the Church in a way that directly helps the Church be the Church of Christ, and thus, Catholic. In everything the person with such a vocation does, they directly represent the Church. (Sometimes they will do so publicly and even officially, other times more privately, but in everything the person is and does, they directly represent the Church.) Moreover, they do so for the sake of the Church; they call directly to other persons within the Church with ecclesial vocations to live their vocations as well and as fully as they can. This is their identity in Christ (another reason we tend to use titles like Sister, Brother, Father, etc., for such persons), and they cannot be this person only some of the time.
This responsibility is about not merely being a Catholic Christian for others, though it includes this, but about representing the Church to herself in ways that allow her to grow to be the Church God calls her to be. Religious are called to witness to and challenge both the laity and clerics in a way that caused John Paul II to comment in Vita Consecrata, that he could not conceive of a Church with only priests and laity (cf Ecclesial Vocations) but without religious. The Church herself recognizes that while religious are not part of the hierarchical nature of the Church (they are not a hierarchical position between clerics and laity), vocationally speaking, they are part of her very holiness. All hermits represent eremitical life in some way, shape, or form. Some of us do this better than others, and some of us do it less well. But canonical hermits are specifically called, and respond in their profession, to both live and explore the vocation in a normative way, aware at every moment that they do so for the sake of God, God's Church, this vocation, and all of those whom this vocation might touch. They are not free to live the life of a hermit in whatever way they want or even in whatever way is comfortable. Canon 603 (for solitary consecrated hermits) and canon and proper law (for those in orders or congregations like the Camaldolese, Carthusians, Carmelites, et al) will dictate and shape the way they live eremitical life. Especially, such hermits will live this life for others' sake -- a phrase that includes all those just noted above.I sincerely hope this answers your questions. You can always get back to me with more questions and comments. Thanks for your patience in awaiting this reply!

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