Reference request: Models of isomorphic languages result into isomorphic categories

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?

• Not time for a full answer now, but a few pointers. Firstly, the specific result I’d look for is the fact that the category of models is functorial in the signature, i.e. any map of signatures $f : \rho \to \sigma$ induces a functor $f^* : \textbf{Str}(\sigma) \to \textbf{Str}(\rho)$, and this action makes $\mathbf{Str}(-)$ a functor from signatures to categories. The isomorphism result then follows immediately. Secondly, two good books to check are Johnstone’s Elephant, and Makkai and Reyes’ First-order Categorical Logic. – Peter LeFanu Lumsdaine Jan 30 '16 at 11:11
• @Peter. I assume that a "map of signatures" is formally a triple $(f,\sigma_1,\sigma_2)$, where $\sigma_i=(\Sigma_{\rm f}^{(i)},\Sigma_{\rm r}^{(i)},\varrho_i)$ is a signature ($i=1,2$) and $f$ is a fnc $\Sigma_{\rm f}^{(1)}\cup\Sigma_{\rm r}^{(1)}\to\Sigma_{\rm f}^{(2)}\cup\Sigma_{\rm r}^{(2)}$ s.t. $f[\Sigma_{\rm f}^{(1)}]\subseteq\Sigma_{\rm f}^{(2)}$, $f[\Sigma_{\rm r}^{(1)}]\subseteq\Sigma_{\rm r}^{(2)}$ and $\varrho_2(f(\zeta))=\varrho_1(\zeta)$ for every $\zeta\in\Sigma_{\rm f}^{(1)}\cup\Sigma_{\rm r}^{(1)}$. If so, I think that I'd rather look for the fact that any morphism of (...) – Salvo Tringali Jan 30 '16 at 13:04
• (...) languages $f: L_1\to L_2$ (whatever this may be...), for which $L_1$ and $L_2$ are languages of type $\mathscr{L}_\infty$, induces a functor $F: {\bf Str}(L_1)\to{\bf Str}(L_2)$. I mean, the result I'm seeking should take into account the axioms that, along with the information provided by the signature, forge the structures modeling the specific language under consideration, right? Thanks anyway for your comment and the pointers (I was already surfing through Johnstone's Elephant, but hadn't thought of Makkai and Reyes). – Salvo Tringali Jan 30 '16 at 13:16
• I'm a little confused by the end of the question. Are you looking for a definition of 'model' or a definition of 'interpretation'? – François G. Dorais Jan 30 '16 at 15:07
• Hájek's article is here: digizeitschriften.de/dms/img/?PID=GDZPPN002043920 You asked for a reference for the fact that "isomorphic languages should result into isomorphic categories [of models]"; Hájek does quite a lot more since he considers translations that aren't just symbol substitutions. I doubt the trivial case you're interested in has been published on its own. – François G. Dorais Jan 30 '16 at 17:39