Fix a category $C$ with finite products and a set $L$ of function symbols (each equipped with an arity in $\mathbb N$).
An $L$-algebra in $C$, $\mathbf A=(A,(f^\mathbf{A})_{f\in L})$, is given by some object $A\in C$ and, for each $n$-ary function symbol $f\in L$, a morphism $f^\mathbf{A}\colon A^n \to A$ in $C$. Using categorical semantics, there is a canonical notion of when an $L$-algebra $\mathbf A$ in $C$ satisfies an $L$-identity $s\approx t$ (given by two $L$-terms $s$ and $t$).
Question: Can one describe the classes of $L$-algebras in $C$ that arise as the class of models of some set of $L$-identities?
Note that Birkhoff's HSP theorem gives an answer to this question in the special case $C=\mathbf{Set}$. If the question is too hard to answer in general, then I would be interested in the case $C=\mathbf{Top}$.
As a bonus question (motivated by the desire to describe the fixed points of the closure operators induced by the Galois connection induced by the satisfaction relation described above): Can one describe the sets of $L$-identities that arise as the set of $L$-identities shared by all $L$-algebras in $C$ in $K$, for some class $K$ of $L$-algebras in $C$? It is a well-known theorem (also due to Birkhoff) that in the case $C=\mathbf{Set}$, these are exactly the sets of $L$-identities closed under the inference rules of equational logic.
