ct.category theory - Image of composite morphisms

This cannot happen in a regular category. Below I give a proof using the sequent calculus of subobjects in a regular category. It can be deciphered using the book 'Sketches of an Elephant Volume 2' by Peter T. Johnstone, in particular chapter D1.

I write $beta:=I$ and $gamma:=J$. I hope the definition of image given in your book is the same as mine, namely the image of a subobject (~mono) $S$ under a morphism $phi$ is the least subobject of the codomain of $phi$ through which $phicircoverline{S}$ factors, where $overline{S}in S$.

Assume we know
$exists x(alpha(x)wedge f(x)=y) dashvvdash_{y:Y}quad beta(y)$ and $exists y(beta(y)wedge g(y)=z) dashvvdash_{z:Z}quad gamma(z)$. We then want to prove two things. The first is that $exists x(alpha(x)wedge g(f(x))=z)vdash_{z:Z}quad gamma(z)$, the second that $gamma(z)vdash_{z:Z} quad exists x(alpha(x)wedge g(f(x))=z)$.

For the first we have the following.
$alpha(x)wedge g(f(x))=z$
$vdash_{x:X,z:Z} quadalpha(x)wedge g(f(x))=z wedge f(x)=f(x)$
$vdash_{x:X,z:Z}quad alpha(x)wedge g(f(x))=z wedge beta(f(x))$
$vdash_{x:X,z:Z}quad gamma(g(f(x)))$. Therefore $alpha(x)wedge g(f(x))=zvdash_{x:X,z:Z}quad gamma(z)$ and hence $exists x(alpha(x)wedge g(f(x))=z)vdash_{z:Z}quad gamma(z)$.

The second also holds. First note that $beta wedge g(y)=z$
$vdash_{y:Y,z:Z}quad exists x(alpha(x)wedge f(x)=y)wedge g(y)=z$
$vdash_{y:Y,z:Z}quad exists x(alpha(x)wedge f(x)=ywedge g(y)=z)$
$vdash_{y:Y,z:Z}quad exists x(alpha(x)wedge g(f(x))=z)$ from which we may conclude that $gamma(z)vdash_{z:Z} quad exists y(beta(y)wedge g(y)=z)vdash_{z:Z} quad exists x(alpha(x)wedge g(f(x))=z)$.