Geodesic
Images in Topoi
Canadian mathematical bulletin, Tome 20 (1977) no. 4, pp. 471-478

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DOI

The construction of images of morphisms in an elementary topos E has hithero required the use of colimits. For example, in [1], Freyd constructs the image of a morphism by taking the equalizer of the cokernel pair of the morphism. In particular, the construction of the direct image functor, or, as it is sometimes referred to, existential quantification along a morphism, has required the use of images, and hence colimits. However, Mikkelsen has defined existential quantification using only limits. 
Rowe, K. A. Images in Topoi. Canadian mathematical bulletin, Tome 20 (1977) no. 4, pp. 471-478. doi: 10.4153/CMB-1977-070-4
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     title = {Images in {Topoi}},
     journal = {Canadian mathematical bulletin},
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     doi = {10.4153/CMB-1977-070-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-070-4/}
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