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For the value at $x$ of the conditional probability density function of a random variable (capital) $X,$ given the event that the value of a random variable (capital) $Y$ is $y,$ I prefer $f_{X\,\mid\,Y\,=\,y}(x)$ to the more frequently seen $f_{X\,\mid\,Y}(x\mid y).$ It emphasizes that we are concerned with a function of $x.$