Twisted arrow category - convention I have found two slight different definitions of twisted arrow category. The part about the arrows in $Tw(C)$. In nCatLab it is something like

given two objects $f:a\to b$ and $f':a'\to b'$ in $Tw(C)$, an arrow from $f$ to $f'$ in $Tw(C)$ is a pair $(g:a\to a',h:b'\to b)$ such that the corresponding diagram commutes.

Barr in his book use something like 

given two objects $f:a\to b$ and $f':a'\to b'$ in $Tw(C)$, an arrow from $f$ to $f'$ in $Tw(C)$ is a pair $(g:a'\to a,h:b\to b')$ such that the corresponding diagram commutes.

That is, $g$ and $h$ in the two definitions have domain and codomain inverted.
Are the consequences of such a difference relevant?
 A: These definitions are dual. 
Let $\mathcal{C}$ be a category. Denote by $\text{Tw}_1(\mathcal{C})$ the category with morphisms of $\mathcal{C}$ as objects and
$$
\hom_{\text{Tw}_1(\mathcal{C})}(f\colon a\to b,f'\colon a'\to b')=\{(g\colon a\to a',h\colon b'\to b)|\quad h\circ f'\circ g=f\}
$$
and by $\text{Tw}_2(\mathcal{C})$ the category with morphisms of $\mathcal{C}$ as objects and
$$
\hom_{\text{Tw}_2(\mathcal{C})}(f\colon a\to b,f'\colon a'\to b')=\{(g\colon a'\to a,h\colon b\to b')|\quad h\circ f\circ g=f'\}.
$$
Then $\text{Tw}_1(\mathcal{C})\cong (\text{Tw}_2(\mathcal{C}))^{\text{op}}$. Indeed, define the functor $\mathcal{F}\colon \text{Tw}_1(\mathcal{C})\to (\text{Tw}_2(\mathcal{C}))^{\text{op}}$ in the following way:
$$
\mathcal{F}_{\text{Obj}}(f)=f;
$$
$$
\mathcal{F}_{\text{Mor}}(g,h)=(g,h)^{\text{op}}.
$$
It is easy to check that $\mathcal{F}$ is an isomorphism of categories.
The more standard definition of twisted arrow category is $\text{Tw}_2(\mathcal{C})$, because $\text{Tw}_2(\mathcal{C})$ is the category of elements of the functor:
$$
\hom_{\mathcal{C}}\colon\mathcal{C}^{\text{op}}\times\mathcal{C}\to\mathbf{Set}.
$$
I didn't find the definition of $\text{Tw}_1(\mathcal{C})$ at the ncatlab (see this), as you pointed in your question.
