The closed sub schemes theoretic image of a morphism of schemes $\phi : X\rightarrow Y $ is defined to be the scheme theoretic intersection of all closed sub schemes containing $\phi $. In case X is reduced or $\phi $ is quasi compact, the closed sub scheme theoretic image of $\phi $ is just the closed sub scheme of Y cut out by the kernels of $\phi^*(U):\mathscr{O}_Y(U)\rightarrow \mathscr{O}_ X(\phi^{-1}(U)) $ where U is an open affine.
My question is whether this is also the image in the category theoretic sense in the category of schemes. If not which properties of the category theoretic image does it have?
At the very least it seems to be a monomorphism since closed sub schemes are monomorphism