Variety in a category

A notion generalizing that of a variety of universal algebras. Let $\mathfrak K$ be a bicategory with products. A full subcategory $\mathfrak M$ of $\mathfrak K$ is called a variety if it satisfies the following conditions: a) if $\mu : A \rightarrow B$ is an admissible monomorphism and $B \in \mathop{\rm Ob} \mathfrak M$, then $A \in \mathop{\rm Ob} \mathfrak M$; b) if $\nu : A \rightarrow B$ is an admissible epimorphism and $A \in \mathop{\rm Ob} \mathfrak M$, then $B \in \mathop{\rm Ob} \mathfrak M$; c) if $A _ {i} \in \mathop{\rm Ob} \mathfrak M$, $i \in I$, then $A = \prod _ {i \in I } A _ {i} \in \mathop{\rm Ob} \mathfrak M$.
If $\mathfrak K$ is a well-powered category, that is, the admissible subobjects of any object form a set, then every variety is a reflective subcategory of $\mathfrak K$. This means that the inclusion functor $I : \mathfrak M \rightarrow \mathfrak K$ has a left adjoint $S : \mathfrak K \rightarrow \mathfrak M$. The unit of this adjunction, the natural transformation $\eta : I _ {\mathfrak K } \rightarrow T = S I$, has the property that for each $a \in \mathop{\rm Ob} {\mathfrak K }$ the morphism $\eta _ {A} : A \rightarrow T ( A)$ is an admissible epimorphism. In many important cases the functor $T$ turns out to be right-exact, that is, it transforms the cokernel $\nu$ of a pair of morphisms $\alpha , \beta : A \rightarrow B$ into the cokernel of the pair of morphisms $T ( \alpha ) , T ( \beta )$, if $( \alpha , \beta )$ is a kernel pair of the morphism $\nu$. Moreover, right exactness and the presence of the natural transformation $\eta : I \rightarrow T$ are characteristic properties of $T$.