According to Wikipedia:
Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category.
I cannot see this for a general category other than the special cases, e.g. $\mathbf{Set}$. Given a general finite product category, we may choose a product object $X \times Y$ of every pair of objects $X$ and $Y$ (which I think is not canonical), a unit object $I$, and the associator \[ \alpha_{X, Y, Z} \colon (X \times Y) \times Z \rightarrow X \times (Y \times Z). \] My question is, how could $\alpha$ be a natural isomorphism, and satisfy the coherence condition?
First, choose a terminal object $1$ and choose for each pair $X, Y$ of objects a product $X \times Y$ (which means we also choose projections $p_0 : X \times Y \to X$ and $p_1 : X \times Y \to Y$). Any choice will work.
For the unitor we require natural isomorphisms $\lambda_X : 1 \times X \to X$ and $\rho_X : X \times 1 \to X$. Well, there is one obvious choice: take the projection $p_2 : 1 \times X \to X$ for $\lambda_X$ and $p_1 : X \times 1 \to X$ for $\rho_X$. I leave it to you to check that these are indeed isomorphisms and natural.
For the associator we require natural isomorphisms $\alpha_{X, Y, Z} : (X \times Y) \times Z \to X \times (Y \times Z)$. Again, there is an obvious choice: take the unique morphism $\alpha_{X, Y, Z}$ such that $p_1 \circ \alpha_{X, Y, Z} = p_1 \circ p_1$ and $p_2 \circ \alpha_{X, Y, Z}$ is the unique morphism such that $p_1 \circ (p_2 \circ \alpha_{X, Y, Z}) = p_2 \circ p_1$ and $p_2 \circ (p_2 \circ \alpha_{X, Y, Z}) = p_2$. Again, I leave it to you check that these are indeed isomorphisms and natural.
Hopefully after you have done the checking yourself you will agree that these are indeed the obvious choices. (There is a precise sense in which these are not only obvious but even forced – there is no other choice that works uniformly!)