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Definition of Exponential object

1. Noun. (category theory) Given objects ''Y'' and ''Z'', the '''exponential object''' $Z^Y$ is uniquely defined by the following universal property: for any object ''X'' with arrow $g:\; X\; \backslash times\; Y\; \backslash rightarrow\; Z$, there can always be constructed an arrow $\backslash lambda\; g:\; X\; \backslash rightarrow\; Z^Y$ which induces an arrow $\backslash lambda\; g\; \backslash times\; \backslash hbox\{id\}\_Y\; =\; i$,   $i:\; X\; \backslash times\; Y\; \backslash rightarrow\; Z^Y\; \backslash times\; Y$ which is unique in satisfying $\backslash text\{eval\}\; \backslash circ\; i\; =\; g$ where $\backslash text\{eval\}:\; Z^Y\; \backslash times\; Y\; \backslash rightarrow\; Z$. ¹

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