It sounds as though what you want is the closure of the image of F under f. (That is, the minimal closed subscheme of Y factoring f.)
If X= Spec A, and Y= Spec B, and F= Spec A/I, and f corresponds to the ring map f′:BtoA, then we can consider the preimage J of I under i′. Consider the set of primes in B containing J. Of course, any prime in the image of F under f must contain J, since its preimage under the ring map has to contain I. Thus, Spec B/J contains the preimage. You can check that it's the biggest such ideal, noting that in order for us to have a map B/JtoA/I, J should be contained in the preimage of I.
Being the closure of the image of F under f is a universal property of sorts (in particular, it's unique), which more or less allows us to argue that this construction generalizes to non-affine schemes. (Just apply this local construction, and uniqueness tells us that the local constructions glue together.)
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