Julio wrote:

> Second, since now we are in \$$\mathcal{C}\$$ (which is any category), if we forget about the hom-functor temporarily and only think about \$$\mathcal{C}\$$, there may well exist various independently defined arrows \$$d \to d'\$$, for \$$d\$$ and \$$d'\$$ are merely two random objects after all. But once we put on the hom-functor spectacles, we are taken into a different (and more restricted) scenery, where the possibly independently existing \$$d \to d'\$$ arrows are no longer important (or even visible), because the hom-functor – which must map/preserve morphisms – needs to establish an _100% secure input-output relation_ in the \$$\mathbf{Set}\$$-perspective between \$$\mathrm{hom}(c,c')\$$ and \$$\mathrm{hom}(d,d')\$$, hence @John's words in the lecture:

>This function should take any morphism \$$h \in \mathrm{hom}(c,c')\$$ and give a morphism in \$$\mathrm{hom}(d,d')\$$.

> Thus, the question is *not* whether there might be \$$d \to d'\$$ arrows in \$$\mathcal{C}\$$ or not (which is a valid question for its own sake but simply uninteresting in our hom-functor discourse), but more restrictively given any \$$c \to c'\$$ arrow as input (together with the relevant morphisms \$$f, g\$$), whether or not we can _confidently guarantee_ at least _one_ such arrow as output.

Right! Well put. You've got it now.

We are looking for a systematic recipe to build a function that takes morphisms in \$$h \in \mathrm{hom}(c,c')\$$ and gives morphisms in \$$\mathrm{hom}(d,d')\$$. We easily get such a recipe if we know morphisms \$$f: d \to c\$$ and \$$g: c' \to d'\$$, and that recipe is what the hom-functor exploits. We don't get such a recipe if we only know morphisms \$$f : c \to d\$$ and \$$g: c' \to d'\$$. So, we need the "op" in

$\text{hom} : \mathcal{C}^\text{op} \times \mathcal{C} \to \mathbf{Set} .$

("Systematic recipe" is vague talk for "functor"; proving that the hom-functor is really a functor, as some students have done above, imposes some constraints that are well-nigh impossible to meet if one isn't systematic.)