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Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far.
Let us write <math>f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)\!</math> to express what has been said so far.
When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the \textit{type} of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its \textit{graph}, letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math>
When it comes to parsing the notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},\!</math> everyone takes the part <math>{}^{\backprime\backprime} X \to Y {}^{\prime\prime}\!</math> to specify the ''type'' of the function, that is, the pair <math>(\mathrm{obj_1}f, \mathrm{obj_2}f),\!</math> but <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> is used equivocally to denote both the triple and the subset <math>\mathrm{obj_{12}}f\!</math> that forms one part of it. One way to resolve the ambiguity is to formalize a distinction between a function and its ''graph'', letting <math>\mathrm{graph}(f) := \mathrm{obj_{12}}f.\!</math>
Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math>
Another tactic treats the whole notation <math>{}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}\!</math> as sufficient denotation for the triple, letting <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> denote <math>\mathrm{graph}(f).\!</math>