Topological abelian groups are additive but not abelian

The category $\mathbf{T}\mathbf{A}\mathbf{b}$ is additive: hom-sets add pointwise, the one-point group is a zero object, and finite products agree with finite coproducts because for finitely many abelian groups the direct product and direct sum carry the same topology.

Let ${\mathbb{R}}_d$ be the additive group of real numbers with the discrete topology and let $\mathbb{R}$ carry its usual topology. The identity homomorphism $\iota :{\mathbb{R}}_d\to \mathbb{R}$ is continuous, bijective, has zero kernel and zero cokernel, so its canonical coimage-to-image map is again $\iota$. But $\iota$ is not an isomorphism in $\mathbf{T}\mathbf{A}\mathbf{b}$, because the inverse map $\mathbb{R}\to {\mathbb{R}}_d$ is not continuous. Hence $\mathbf{T}\mathbf{A}\mathbf{b}$ is additive but not abelian.