There is a construction of Peter Freyd that embeds any triangulated category $\mathcal{T}$ in an abelian category $\mathcal{A}(\mathcal{T})$. Explicitly, we start with the category of arrows in $\mathcal{T}$. Given a morphism $u$ in $\mathcal{T}$, I'll write $I(u)$ for the same thing regarded as an object of the arrow category. Next, we identify two morphisms $(f,g):I(u)\to I(v)$ and $(f',g'):I(u)\to I(v)$ if the diagonal composite $vf=gu$ is the same as the diagonal composite $vf'=g'u$. It is straightforward to see that this gives a quotient category, which we call $\mathcal{A}(\mathcal{T})$. It is quite a long and interesting argument to show that this is actually an abelian category. It is almost never well-powered unless $\mathcal{T}$ is small.

There is a construction of Peter Freyd that embeds any triangulated category $\mathcal{T}$ in an abelian category $\mathcal{A}(\mathcal{T})$. Explicitly, we start with the category of arrows in $\mathcal{T}$. Given a morphism $u$ in $\mathcal{T}$, I'll write $I(u)$ for the same thing regarded as an object of the arrow category. Next, we identify two morphisms $(f,g):I(u)\to I(v)$ and $(f',g'):I(u)\to I(v)$ if the diagonal composite $vf=gu$ is the same as the diagonal composite $vf'=g'u$. It is straightforward to see that this gives a quotient category, which we call $\mathcal{A}(\mathcal{T})$. It is quite a long and interesting argument to show that this is actually an abelian category. It is almost never well-powered unless $\mathcal{T}$ is small.

This is from the "Proceedings of the Conference on Categorical Algebra" held in La Jolla in 1965, published by Springer.