The associativity condition in a strict monoidal category is very often defined just at the object level, namely, for objects $x,y,z$ in a strict monoidal category $\mathsf{C}$, one has:

$$(x\otimes y)\otimes z = x\otimes (y\otimes z)$$

Does the associativity at morphism level derive from this condition or is it a part of the definition. As I see, for $f:x\to x'$, $g:y\to y'$ and $h:z\to z'$ the two morphisms $(f\otimes g)\otimes h$ and $f \otimes (g\otimes h)$ share the same source and target objects but are not necessarily the same (what really does *the same* mean in strict case!).