Concrete full embeddings into categories of algebras and coalgebras (Q1321040)

A full embedding \({F}\) from a concrete category \({\mathfrak K}\) into a concrete category \({\mathfrak L}\) is called a realization if \({F}\) preserves underlying sets and mappings. For set functors \({F}\), \({G}\) define a category \({\mathfrak A}(F,G)\) whose objects are pairs \((X\), \(\varphi\colon FX\to GX)\) and morphisms from \((X\), \(\varphi\colon FX\to GX)\) to \((Y\), \(\psi\colon FY\to GY)\) are all mappings \(f\colon X\to Y\) with \(Gf\circ \varphi= \psi\circ Ff\). A concrete category \({\mathfrak K}\) has a realization in \({\mathfrak A}(F,G)\) for some set functors \({F}\) and \({G}\) if and only if \({\mathfrak K}\) is strongly small fibred and satisfies the zig-zag condition.