According to the definition of arrow category at nlab website, if category C has $a_0,a_1,b_0,b_1$ as objects and following morphisms:
$a:a_0 \rightarrow a_1$
$b:b_0 \rightarrow b_1$
$f:a_0 \rightarrow b_0$
$g:a_1 \rightarrow b_1$
then the arrow category Arr(C) will have the following morphism:
$<f,g>: <a_0,a,a_1> \rightarrow <b_0,b,b_1>$
But what would happen if instead we had following morphisms in C:
$a:a_0 \rightarrow a_1$
$b:b_0 \rightarrow b_1$
$f:a_0 \rightarrow b_0$
or if having following:
$a:a_0 \rightarrow a_1$
$b:b_0 \rightarrow b_1$
$f:a_0 \rightarrow b_0$
$g:b_1 \rightarrow a_1$
How would those setups be represented in arrow category?