Let \$$\mathbf{Matrix}\$$ be a category whose objects are matrices and morphisms are matrix homomorphisms.

Does there exist a \$$F,U\in \mathrm{Mor}(\mathbf{Matrix})\$$ such that,
\$\left( \begin{array}{cc} 1 & 1 \\\\ 1 & 0 \end{array} \right) \overset{F}{\rightarrow} \left( \begin{array}{cc} f & g \\\\ h & 0 \end{array} \right) \$
and,
\$\left( \begin{array}{cc} f & g \\\\ h & 0 \end{array} \right) \overset{U}{\rightarrow} \left( \begin{array}{cc} 1 & 1 \\\\ 1 & 0 \end{array} \right) ? \$