TITLE: Proof that a Cartesian category is monoidal
QUESTION [15 upvotes]: If $\mathcal C$ is a category with products and a terminal object, then $\mathcal C$ is monoidal.
This seems obvious, but wherever I look for a proof or a reference it simply states that the proof is "pretty involved".
I found this 8 years old question on MathOverflow:
Monoidal structure on a category with products and with terminal object
The answer given there uses the fact that Set is monoidal, along with the Yoneda lemma.
Is there any other reference that proves it more "generally" (i.e without resorting to Set), or a standard reference?

REPLY [3 votes]: One can find a completely explicit, constructive proof of this in the agda-categories library too, with explicit equational proofs for all pieces. There's an html version for easy web readability too (go to the almost bottom of that file).  The symmetric monoidal category is worked out too.
The advantage of doing it this way is that, if you ever wonder why some specific detail is true, it's all here to be seen. No details omitted.