This is pretty loosey-goosey, but points to a very real phenomenon. For instance, a Lawvere theory is a category with finite products, and a model in sets is a finite product-preserving functor into the category of sets; a model of the Lawvere theory given by the opposite of the category of finitely generated free groups is identified with a group in the traditional sense. But this needs finite product preserving functors, and many similar categorical versions of universal algebra have similar restrictions. For a more direct example of what is described here, one can view a monoid $M$ as a one-object category $BM$, and it is then reasonable to view $BM$ as the "theory of $M$-actions." A functor from $BM$ to the category of sets is identified with an $M$-set, and this gives a way of interpreting the theory in a completely arbitrary category. Concretely, this amounts to saying that an action of $M$ on an object $x$ in some category is a monoid homomorphism from $M$ into the endomorphism monoid of $x$.