Injective objects in topoi.

Dear Category theorists, I'm sure the following lemma is either known, or I've made some mistake in proving it: Lemma: If I is an injective object in a topos, then for any monomorphism I>---m--->A, the object of splittings of m, i.e. S={f:I^A|(\forall i:I)(f(m(i))=i)} is also injective. Could someone let me know of a reference to it. Thanks a lot. Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]