There is no slack, it's under tension! Nominally the string is in a straight line between nut and bridge, only nominally because we are ignoring gravity effects. For it to be bent to contact the fret it has to be physically longer and for it to be longer it must stretch and for it to be stretched it must be at more tension than when it was straight. The deviation from straight is small so the increase in tension is also small, but it is inescapable.
The force that is required to push the string over to the fret is amplified manyfold in the increased string tension, by the ratio of the string length to the deflection.