Assigning a profile ...

Question

I think that there are two different relationships between an image and a color space ... embedding and assigning.
What is the difference between these two things?
There are no shortage of articles on the Internet trying to explain the subject by some analogies, but I could not see any direct response and could not understand ... maybe, because it is a little bit late time here now )
Thanks a lot ...

Answer 1

Assigning a profile is a process used to provide it's (hopefully correct) color space. Once done, that profile is embedded into the document.
Here's my very old analogy of why we need to assign profiles (and embed them):
Here’s a metaphor for a color space: suppose I supply a recipe for chocolate chip cookies but do not provide the unit for each ingredient in the recipe. The recipe provides each ingredient followed by a number. Without units you can’t make the cookies. The numbers alone are not enough information to describe how the cookies will be produced. Likewise, R78/G103/B23 or C23/M98/M123/K6 is not enough information to reproduce that color.
Going back to the chocolate chip cookie analogy, suppose a color model is a cookie recipe with only three ingredients. I give you this recipe, which simply calls for 1-flour, 8-butter and 2-chocolate chips. You don’t have enough information to make the cookies. However if I provide you the recipe with a specific scale—1 cup of flour, 8 tablespoons of butter, and 2 cups of chocolate chips—I’ve provided the necessary information, the scale, to make a dozen chocolate chip cookies. I can give you the cookie recipe in the metric scale such as liters and grams and you can still makes the same cookies even though the numbers are different. A color space is a color model that has a known reference and scale, in this case primaries (the ingredients, in this case RG and B) and scale (specific quantities of these ingredients).
Suppose I specify a color as R10/G130/B50 and specify a color reference by saying the color space is Adobe RGB (1998), which defines the scale of the RGB primaries; the color coordinates of this color space. The R10/G130/B50 set of numbers can now reproduce a color by anyone with the proper tools since the reference and scale have been defined. Different RGB color spaces use a different scale of red, green, and blue primaries. Adobe RGB (1998) and sRGB are different color spaces, however both are based on the RGB color model using RGB primaries. Although each color space uses the same three primary ingredients (R, G, and B), the specific colorimetric scale of each color space is different. The maximum of red, green, and blue are more saturated in the Adobe RGB (1998) color space than the sRGB color space. Even though R0/G255/B0 is the greenest green ingredient in both Adobe RGB (1998) and sRGB, knowing that the scale is different in both color spaces explains why this green value is more saturated in Adobe RGB (1998). This also illustrates how R0/G255/B0 alone can’t tell us what green.
An ICC profile simply defines this scale and gives the numbers a meaning allowing us to reproduce the color using something far more concrete than using the English word "Green" or a set of numbers which alone is far too ambiguous to produce a specific color appearance.
So R255 in Adobe RGB (1998) and R255 in sRGB share the same numbers but have a different scale. When you assign a color space to either number, you’re telling Photoshop the scale, it’s updating it’s preview and showing you this (correct or not, you’ve defined the scale so Photoshop believes you). The number hasn’t changed the color appearance has. Now you know why.