>>16771402
No, this implies excluded middle. The inclusion [math]\iota:\{0\mid \sigma\}\hookrightarrow\{0\mid\sigma\}\cup\{1\}[/math] is constant in the sense that [math]\iota(x)=\iota(x')[/math] for every [math]x,x'[/math] in its domain and its codomain is clearly inhabited. But if there was a [math]y[/math] in its codomain such that [math]\iota(x)=y[/math] for every [math]x[/math] in its domain then [math]\sigma\lor\neg\sigma[/math] must hold, for either [math]y\in\{0\mid\sigma\}[/math] in which case [math]\sigma[/math] holds and we're done, or [math]y=1[/math]. But in the [math]y=1[/math] case [math]\neg\sigma[/math] must hold, since if [math]\sigma[/math] was true we'd have [math]0\in\{0\mid\sigma\}[/math] and so [math]\iota(0)=y=1[/math], but [math]\iota(0)=0[/math] by definition of [math]\iota[/math], so [math]0=1[/math], contradiction.

The answer (though given without proof) was also just a quick google search away (pic rel taken from https://ncatlab.org/nlab/show/constant+function)...
But at least those are more interesting questions than whatever the Euclid schizo is talking about.