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Charles Ehresmann defined in 1963 a double category
 as an internal category in the category of small categories  .
Definition 0.1 A double category
 consists of:
- a set of objects,
- a set of horizontal morphisms
- a set of vertical morphisms
and
- a class of squares with source and target as shown in the following diagrams:
with compositions and units of the double category that satisfy the following axioms:
- i. Horizontal:
- ii. Vertical:
Compositions for square diagrams in a double category
 :
- iii. Horizontal composition:
- iv. Vertical composition of squares in
 :
![${[\alpha \beta]}_{vert.}$](http://images.physicslibrary.org/cache/objects/761/l2h/img14.png "${[\alpha \beta]}_{vert.}$") is expressed as
Moreover, all compositions are associative and unital, and also subject to the Interchange Law:
Unit morphisms are also subject to the axioms of the double category. For further details on double categories and examples please see the related free download PDF file.
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"double category" is owned by bci1.(view preamble)
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\ {C}\ar[r]_{h}&{D} } } \end{xy}$](http://images.physicslibrary.org/cache/objects/761/l2h/img6.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {A}\ar[r]^{f}\ar[d]_{k}&{B}\ar[d]^{g}\ {C}\ar[r]_{h}&{D} } } \end{xy}$")
![$\displaystyle A\buildrel f_1 \over \longrightarrow B \buildrel f_2 \over \longrightarrow C = [f_1, f_2]= f_2 \circ f_1 $](http://images.physicslibrary.org/cache/objects/761/l2h/img7.png "$\displaystyle A\buildrel f_1 \over \longrightarrow B \buildrel f_2 \over \longrightarrow C = [f_1, f_2]= f_2 \circ f_1 $")
![$\displaystyle [A\buildrel j_1 \over \longrightarrow B \buildrel j_2 \over \longrightarrow C]_{vert} = [j_1, j_2]_{vert.}= j_2 \circ j_1 $](http://images.physicslibrary.org/cache/objects/761/l2h/img9.png "$\displaystyle [A\buildrel j_1 \over \longrightarrow B \buildrel j_2 \over \longrightarrow C]_{vert} = [j_1, j_2]_{vert.}= j_2 \circ j_1 $")
![$\displaystyle [A\buildrel 1^v_A \over \longrightarrow A \buildrel j_1 \over \lo... ...l j_1 \over \longrightarrow B \buildrel 1^v_B \over \longrightarrow B]_{vert.} $](http://images.physicslibrary.org/cache/objects/761/l2h/img10.png "$\displaystyle [A\buildrel 1^v_A \over \longrightarrow A \buildrel j_1 \over \lo... ...l j_1 \over \longrightarrow B \buildrel 1^v_B \over \longrightarrow B]_{vert.} $")
![$\displaystyle \xymatrix{ {A}\ar[r]^{f_1}\ar[d]_{j}&{B}\ar[d]^{k}\ {D}\ar[r]_{... ...f_1f_2]}\ar[d]_{j}&{C}\ar[d]^{l}\ {D}\ar[r]_{g_1g_2}&{F}} ~~~~[\alpha \beta].$](http://images.physicslibrary.org/cache/objects/761/l2h/img12.png "$\displaystyle \xymatrix{ {A}\ar[r]^{f_1}\ar[d]_{j}&{B}\ar[d]^{k}\ {D}\ar[r]_{... ...f_1f_2]}\ar[d]_{j}&{C}\ar[d]^{l}\ {D}\ar[r]_{g_1g_2}&{F}} ~~~~[\alpha \beta].$")
![$\displaystyle \xymatrix{ {A}\ar[r]^{f}\ar[d]_{[j_1 j_2]_v}&{B}\ar[d]^{[k_1 k_2]_v}\ {E}\ar[r]_{h}&{F}}~~~~[\alpha \beta]_v.$](http://images.physicslibrary.org/cache/objects/761/l2h/img15.png "$\displaystyle \xymatrix{ {A}\ar[r]^{f}\ar[d]_{[j_1 j_2]_v}&{B}\ar[d]^{[k_1 k_2]_v}\ {E}\ar[r]_{h}&{F}}~~~~[\alpha \beta]_v.$")
![$\displaystyle \xymatrix{ {[\alpha]}\ar[r]^{--}\ar[d]_{\vert}&{[\beta]}\ar[d]^{\... ... ~~over~~ [\gamma \delta]]}_{vert.} = [\alpha \gamma]_v \circ [\beta \delta]_v.$](http://images.physicslibrary.org/cache/objects/761/l2h/img16.png "$\displaystyle \xymatrix{ {[\alpha]}\ar[r]^{--}\ar[d]_{\vert}&{[\beta]}\ar[d]^{\... ... ~~over~~ [\gamma \delta]]}_{vert.} = [\alpha \gamma]_v \circ [\beta \delta]_v.$")