defining a strictly associative product By a category with finite products, I mean a category $ \mathcal{C} $ with a specified terminal object $ 1 $ and a specified binary operation $ \_ \times \_  $ on the objects of $ \mathcal{C} $ so that $ X \times Y $ is the product of $ X $ and $ Y $ along with specified projection maps. I assume that this allows one to uniquely define an operation which for all $ n$ maps an $n$-tuple of objects $X_1, \ldots,X_n$  to $ X_1 \times \ldots \times X_n := ((X_1 \times X_2) \times \ldots \times X_{n-1} ) \times X_n $ and also associates the corresponding $n$-ary projection maps $ \pi_1, \ldots \pi_n $.
My question is the following: Is it always possible to specify another product "operation" $ \otimes $ on $ \mathcal{C} $ such that whenever $ X = X_1 \otimes \ldots \otimes X_n $ and $ Y = Y_1 \otimes \ldots \otimes Y_m $, then $ X \otimes Y = X_1 \otimes \ldots \otimes X_n \otimes Y_1 \otimes \ldots \otimes Y_m $. I think put more succinctly my question is: If you can define a $n$-ary products in a category, can you define a product opertion in the category which is strictly associative?
So I think my question is very similar to enter link description here. There I think the question is about coproducts instead of products and the author is asking whether it is possible to define coproducts in the particular category $ Set $ such that for every $ 3 $-tuple of sets one has strict associativity. Is this true? I am also a little unclear about the solution which seems to say this is not possible as I don't know much set theory.
Finally, my motivation for asking this question is that I am interested in categorical model theory. There typed $n$-ary operation symbols $ f \colon \sigma_1, \ldots \sigma_n \to \tau $ are interpreted in a category with finite products as a morphism $ [[f]] \colon X_1 \times \ldots \times X_n \to Y $. Now one can add extra types such as $ \sigma_1 \times \sigma_2 $ to be interpreted as a product and corresponding operations on the morphism symbols to be products. Now when interpreting these operation symbols as morphisms via composition, if we don't have strict associativity it seems you have to add all these maddening isomorphism maps to guarantee  that things compose and it is often not clear to me that things work out as intendend. I often think I am failing in some understanding about category theory because it seems like from what I've read these difficulties should be trivial.
Thank you very much for your time.
 A: I'll respond to your motivating problem first.
Every category with finite products $\mathbf C$ is equivalent to a category $\mathbf C'$ with strictly associative and strictly unital finite products (this follows from the coherence theorem for monoidal categories). We can always interpret syntactic operations strictly in $\mathbf C'$: the syntactic reasoning may then be transferred across the equivalence $\mathbf C' \to \mathbf C$ (noting that equivalences preserve finite products). It's therefore unnecessary to consider the structural isomorphisms explicitly.
Finally, note that it is not enough just for $(X \times Y) \times Z = X \times (Y \times Z)$. We also want the structural isomorphism $\langle\pi_1\pi_1, \langle \pi_2\pi_1, \pi_2\rangle\rangle : (X \times Y) \times Z \to X \times (Y \times Z)$ to be the identity (which implies the two objects are equal). (And similarly for the units $\pi_1 : X \times 1 \to X$ and $\pi_2 : 1 \times X \to X$.)
In response to the question in your title, the answer is no. A counterexample is the skeleton of the category of sets, as proven in Proposition 7.5 of Kock's 1967 thesis Limit monads in categories.
