The paper investigates the connection of three topics – the categories considered as algebras over graphs, implicit operations describing pseudovarieties, and profinite monoids and categories. Denote by \({\mathcal C}\) the \({\mathcal C}\)-variety of all finite categories. First, the author characterizes profinite categories and constructs the \({\mathcal V}\)-profinite completion \(\widehat C_{\mathcal V}\) of a finite vertex category \(C\) for a \({\mathcal C}\)- variety \({\mathcal V}\). If \(X^*\) is a free category over a graph \(X\), then \(\widehat X^*_{\mathcal V}\), called a pseudofree category over \({\mathcal V}\) on \(X\), is free in the category of all \({\mathcal V}\)-profinite categories. The notions of content, support, and bonded normal form defined on \(X^*\) can be extended to \(\widehat X^*_{\mathcal C}\) and to “most” \(\widehat X^*_{\mathcal V}\). Any nontrivial \({\mathcal C}\)-variety is a class of finite categories determined by bonded pseudo-identities. If \({\mathcal N}\) is a monoid pseudovariety, denote by \(g{\mathcal N}\) and \(\ell {\mathcal N}\) the smallest or the largest \({\mathcal C}\)-variety associated with \({\mathcal N}\), then the pseudo-identities determining \(g{\mathcal N}\) or \(\ell {\mathcal N}\) are derived in terms of pseudo-identities determining \({\mathcal N}\). Denote by \(\overline {\Omega_X {\mathcal V}}\) the category of all implicit operations over \({\mathcal V}\) on \(X\). Then there exists an isomorphism from \(\widehat X^*_{\mathcal V}\) into \(\overline {\Omega_X {\mathcal V}}\) which is a homeomorphism. An application for monoid pseudovarieties and for classes of languages recognized by graphs, and a generalization to semigroupoids (i.e., categories without identity) are derived.