How is the notation $f \times A$ used? I am a non-mathematician, with some working knowledge of category theory, though.  Reading Riley's Catgories of Optics, I found some strange notation I can't remember being introduced anywhere. For example, page 4 defines:

Given two pairs of objects of $\mathcal{C}$, say $(S, S')$ and  $(A, A')$, an optic $p : (S, S') \to (A, A')$ is an element of the set of pairs $(l, r)$, where $l : S \to M \otimes A$ and $r : M \otimes A' \to S'$, quotiented by the equivalence relation generated by relations of the form
$$ ((f \otimes A) l, r) \sim (l, r(f \otimes A')) $$
for any $l : S \to M \otimes A$, $r : N \otimes A' \to S'$, and $f: M \to N$.

Reading this $f \otimes A$ in the relation, my inner type checker complained: you cannot simply pair a morphism with an object, can you? (I know what a product of morphisms is, but this can't be one.)
Intuitively, I assumed this to be a typo for $f \otimes \operatorname{id}_A$, the only thing that would make sense to me.  But some pages before, a morphism $$T \times A \xrightarrow{[\operatorname{id}_T, \operatorname{GET}_1] \times A} T \times S \times A$$  is used -- which makes it obvious that this is not the same thing, and this seems to be a notational convention.
Now, notation is always difficult to search for. The closest matching example of this was in Loregian's Coend calculus, who, for example on page 2, liberally puts morphisms in places where objects are expected: $X^B \times f$, $\operatorname{Set}(A, X^f)$, and others.
From the associated diagrams, I came to the believe that this is intended to mean the application of a functor with a hole in the place of $f$ to the morphism: $(X^B \times -)(f)$, with the other objects fixed.
But this still left me wondering what the exact usage is.  Is there really always a unique functor for "composite objects with holes"?  Wouldn't $f \times \phi_A$  for any idempotent $\phi_A$ also work instead of $f \times \operatorname{id}_A$, for example?  Or is it convention to mean the "natural thing"?  Then, how exactly can $\operatorname{id}_A$ be called more natual than $\phi_A$?
Or is it just to differentiate "$f \times \operatorname{id}_A$ are accidentially being paired here" from "this is an application of the functor $- \times A$!"?
 A: This notation represents fixing an object in a functor with multiple arguments.
Let $F : \mathbf C \times \mathbf D \to \mathbf E$ be a bifunctor (i.e. a functor from a product category), and let $C \in \mathbf C$ be an object. Then $F(C, -) : \mathbf D \to \mathbf E$ is also a functor, defined by precomposing $F$ with $\langle \underline C, \mathrm{Id}_{\mathbf D}\rangle : \mathbf D \to \mathbf C \times \mathbf D$, where $\underline C$ is the constant functor that picks out $C$.
As a functor, $F(C, -)$ acts on objects and on morphisms, so we can write $F(C, D)$ or $F(C, f)$. Technically, this is a different functor from our original $F$, but note that the new $F(C, f)$ is given by the old $F(\mathrm{id}_C, f)$, so the notational convention for fixing an argument matches picking the identity morphism for the fixed object, and so the notation is unambiguous.
In particular, tensor products in monoidal categories are given by bifunctors $(-) \otimes (-) : \mathbf C \times \mathbf C \to \mathbf C$, which includes the cartesian product $(\times)$, and so we may fix an object in either argument to form a functor $\mathbf C \to \mathbf C$.
I haven't seen this convention explicitly mentioned in any introductory texts, but it is very common (as you've noticed), so it is worth being aware of.
