Love and Responsibility goes into great depth about the difference between Comradeship and Friendship; so much so, that I think Comradeship deserves its own formula as does Karol Wojtyla! He emphasizes the objective qualifiers in Comradeship as prerequisites for friendship. In other words, somewhat like sympathy, comradeship is a great aid to friendship. I would attempt to formulate it like this, with explanation to follow:
ᵾ Work or ᵾ Company + ᴲ individual’s role at work or company → ᵾ public ᴲ identity with both ᴲ role & ᵾ Company
Wojtyla’s argument for the value in comradery is that it fosters community; it essentially brings the existential into the universal. While not sacramental, like a marriage involving the friendship between a man and a woman, comradeship does allow for a kind of stewardship of family in the broader environment of neighborhood and city, etc. Simply put, it brings people together for a common purpose akin to solidarity.
In other sections of Love and Responsibility, he calls the process of fostering community “integrating” the exclusive love between man and woman into the broader network of relationships. He even warns that if a couple fails to integrate; their own relationship may not succeed. In terms of comradery then, it can determine the success or failure of a friendship.
Seen side by side, the two formulas (friendship and comradeship) look like this. *Please note, sympathy as defined by Wojtyla simply means "shared life together":
ᴲ Sympathy→ Conformity of Wills- ᵾ -( ᴲ Trust & ᴲ Sympathy, ᴲ Forgiveness & ᵾ Reconciliation) → ᴲ Virtue of Hope ᵾ (ᴲ Trust & ᴲ Sympathy) = Friendship
ᵾ Work or ᵾ Company + ᴲ individual’s role at work or company → ᵾ public ᴲ identity with both ᴲ role & ᵾ Company = Comradeship
Integrated together, I would say they look like this (where “Comradeship” respresents its own formula in the form of just the single word) :
ᴲ Sympathy→ Conformity of Wills- ᵾ -( Comradeship ᵾ, ᴲ Trust & ᴲ Sympathy, ᴲ Forgiveness & ᵾ Reconciliation) → ᴲ Virtue of Hope ᵾ (ᴲ Trust & ᴲ Sympathy) = Friendship
I added “Comradeship” to the “Conformity of Wills” section of the Friendship formula on account of it being integral to function of community. Even St. Paul refers to the “proper functioning of each part” of the Body of Christ, and if certain parts compete or are envious of each other, then functioning is disintegrated.
 Wojtyla, Karol. Love and Responsibility. “From Sympathy to Friendship”. P. 94 Ignatius: San Francisco, 1993
 Wojtyla, Karol. Love and Responsibility. “The Problem of Integrating Love”. P. 114-118 Ignatius: San Francisco, 1993
________________________________________________________Fr. John Nepil has a thorough article on the quality of certain friendships: pleasant or true, according to Aristotle and the process of either making beer or scotch. I found it interesting in that the chief distinguisher between a true friendship and a pleasant one is reconciliation. In other words, true friendship—like the intense process of making scotch (chemical change and transformation after distilling) —requires painful and transforming demands of forgiveness and reconciliation. His approach adds wholly new (ᴲ) existential and (ᵾ) universal qualifiers to my formula for friendship derived from Karol Wojtyla:
ᴲ Sympathy→ Conformity of Wills- ᵾ -(ᴲ Trust & ᴲ Sympathy, ᴲ Forgiveness & ᵾ Reconciliation) → ᴲ Virtue of Hope ᵾ (ᴲ Trust & ᴲ Sympathy)