This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory.

Fix an uncountable universe $\mathscr{U}$ within Tarski-Grothendieck set theory, and denote by $\mathcal C^+$ the set of all non-zero $\mathscr U$-small cardinals and by $\mathscr L_\infty$ the usual extension of the first-order logic, where we allow for simultaneous quantification over fewer than $\lambda$ variables, as well as for conjunctions and disjunctions of size less than $\kappa$, under the proviso that $\kappa, \lambda \in \mathcal C^+$.

We let a ($\mathscr{U}$-small, single-sorted) signature be a triple $\sigma = (\Sigma_{\rm f}, \Sigma_{\rm r}, \varrho)$ consisting of $\mathscr{U}$-small sets $\Sigma_{\rm f}$ and $\Sigma_{\rm r}$, whose elements are labeled, respectively, the function and relation symbols of the signature, and a function $\varrho: \Sigma_{\rm f} \cup \Sigma_{\rm r} \to \mathcal C^+$ assigning an ariety to each (function or relation) symbol, with the condition that $\Sigma_{\rm f}$ and $\Sigma_{\rm r}$ are disjoint from each other and from the set of (logical and non-logical) symbols of $\mathscr{L}_\infty$.

I would like to make formal sense of the (naive) idea that the actual symbols in $\Sigma_{\rm f}$ and $\Sigma_{\rm r}$ are irrelevant modulo the kind of issues emphasized, e.g., on p. 1 of W. Hodges' Model Theory. More precisely, I'm referring to the concept that "the $\mathscr{U}$-small models of isomorphic languages should result into isomorphic categories", where the term 'model' is to be understood in the sense of model theory: I'm putting the text between quotation marks because I'm actually looking for a place in the literature where it is made into a formal statement (I think I know how to do it myself, but that's not the point here), or at least discussed to a satisfactory degree of detail. So my question is: Where can I find anything along these lines?