According to Ockham's first theory, in what way do 'universals' exist?

In each case, consider a universal to be abstract. This gets
complicated because some nominalists (or conceptualists) might consider numbers to be real in the
abstract but something like redness does not. For the sake of argument, let's say that universals
are abstract and therefore, do not exist by themselves; they only exist as exemplified by
particular, real objects. Thus, strength does not exist abstractly by itself as universal
'strength.' But particular, real objects like lions can exhibit strength. Since each exhibition
of strength depends on a particular object/thing, it is individual; not universal. So, for
Ockham, universals are abstract but does this mean they don't exist at
all?

It depends. Ockham is either considered a nominalist, a
conceptualist or a terminist in the beginning of his philosophy. All three are similar but
slightly different. As a nominalist, Ockham would believe that universals only exist as names
(mentally, spoken or written.) 'Humanity' exists only as a word. Humanity in the real world is
different for each human. There is no real universal humanity. It is only an abstract name(word)
that, like strength, can be exhibited individually each human.

As a
conceptualist, Ockham would believe that a universal existed as a mental concept. In a sense,
this means that a universal "stood for" or "represented" the Real object - only in thought. So,
'redness' or 'humanity' are mentally produced concepts which reflect a MENTAL
common/universal ontology shared by all things red or human. They are concepts produced when
supposing their existence. So, they are a bit more than just words (names). This is just a tricky
way of saying these concepts are boiled down to the terms we come up with when trying to think of
universals. Thus, he is also described as a terminist. The red things and human things in the
real world do not share a universal or essential redness or humanity. As a terminist or
conceptualist, universals like redness, humanity and strength are just metaphysical concepts or
terms produced by the act of thinking of all the things that INDIVIDUALLY
EXHIBIT these universals.

So, kind of bridging the
three (nominal, conceptual and terminist), Ockham's first theory of universals could be
considered a phenomenological theory of thought-intentionality; MEANING THAT
UNIVERSALS ARE MENTAL CONSTRUCTS - they are metaphysical/abstract but not real. In
other words, Ockham believed universals to be products of the actual process of mentally
considering universals. Redness is a term that you mentally consider when you think about things
that exhibit redness. Redness is a property of a red thing, but redness, by itself, exists only
as a mental consideration. There is no essential 'redness" somewhere out
there.