nLab
connection on a double category

Let $D=(D_1,D_0)$ be a double category. A connection on $D$ is given by a pair of functions $\Gamma ,\Gamma \prime :\mathrm{arr}{D}_0\to \mathrm{arr}{D}_1$, that assign to each vertical morphism $f:Y\to X$ cells of the form

$$\begin{array}{ccc}Y& \stackrel{f^{*}X}{\to }& X\\ f\downarrow & \Gamma f\Downarrow & \downarrow 1\\ X& \underset{{\iota }_X}{\to }& X\end{array}\begin{array}{ccc}Y& \stackrel{{\iota }_Y}{\to }& Y\\ 1\downarrow & \Downarrow \Gamma \prime f& \downarrow f\\ Y& \underset{f^{*}X}{\to }& X\end{array}$$\begin{matrix} Y & \overset{f^*X}{\to} & X \\ \mathllap{f} \downarrow & \mathllap{\Gamma f} \Downarrow & \downarrow \mathrlap{1} \\ X & \underset{\iota_X}{\to} & X \end{matrix} \qquad \qquad \begin{matrix} Y & \overset{\iota_Y}{\to} & Y \\ \mathllap{1} \downarrow & \Downarrow \mathrlap{\Gamma' f} & \downarrow \mathrlap{f} \\ Y & \underset{f^*X}{\to} & X \end{matrix}

(where ${\iota }_X$ is the horizontal identity on $X$) such that $\Gamma$ and $\Gamma \prime$ behave suitably with respect to composition and identities in $D_0$:

…

(that is, they are identity-on-objects functors) and such that the vertical composite $\Gamma \prime (f)\cdot \Gamma (f)={\iota }_f$ and the horizontal composite $\Gamma (f)\circ \Gamma \prime (f)=1_{f^{*}X}$, (that is, $f$ and $f^{*}X$ are companions).