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Here is a low-level observation, which I think I read on a different MO thread somewhere. If the objects in your category behave like the category of sets, then a map into an object $X$ can be "local" (its image might be a small subobject), but a map out of $X$ must always be "global" (it has to be defined on all of $X$). So in some sense even a single map out of $X$ (e.g. a Moore Morse function in the category of smooth manifolds) can capture much more of the structure of $X$ than a single map into $X$.

Of course if your category behaves like the opposite of the category of sets then the opposite is true. And by the Yoneda lemma both maps into an object and maps out of an object classify it up to isomorphism, so I don't think it necessarily makes sense to privilege either point of view in general.

There is some really interesting general discussion of these issues in Lawvere and Schanuel's Conceptual Mathematics.

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Here is a low-level observation, which I think I read on a different MO thread somewhere. If the objects in your category behave like the category of sets, then a map into an object $X$ can be "local" (its image might be a small subobject), but a map out of $X$ must always be "global" (it has to be defined on all of $X$). So in some sense even a single map out of $X$ (e.g. a Moore function in the category of smooth manifolds) can capture much more of the structure of $X$ than a single map into $X$.

Of course if your category behaves like the opposite of the category of sets then the opposite is true. And by the Yoneda lemma both maps into an object and maps out of an object classify it up to isomorphism, so I don't think it necessarily makes sense to privilege either point of view in general.