I am wondering if there is a standard definition for a category with objects as first order logical (FOL) expressions e.g. $\neg x \vee y$. It seems to me that these logical expressions would be part of a concrete category as they can be mapped (forgetfully) to sets, and as many concrete categories (groups etc.) have homomorphisms as morphisms that perhaps homomorphisms are the morphisms in this FOL category. I am having difficulties visualizing what these homomorphisms would look like - perhaps just some operation $ (\circ,\bullet) $ such that
$f((x \wedge y) \vee z) \rightarrow (f(x) \bullet f(y)) \circ f(z)$ where $(\vee,\wedge) \rightarrow (\circ,\bullet)$ ?